I am looking for a photo or chart of the 28 (I Think) solutions for a standard 16 square, Magic Square. ( Vertical, Horizontal, 4 Corners, Diagonals, etc etc )

I usually hand out the completed Magic Square and want to include another sheet with a chart of all the combinations.

PMed you.

Hi,

Download the paper on the page

http://www.ehu.es/~mtpalezp/mates/lamat.pdf

and look at the pages 9 and 10. There you'll find more than 28 solutions of the magic square.


Posted: Oct 26, 2010 1:01pm
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I forgot to mention that the paper is written in Spanish but I think you can understand it well.

Hello GALIER,

Yes, that is the type of chart I am looking for. Although I want to convert the numbers shown into just color squares to show the variations.

Thank You

Wow, Galier, thank you for this reference.

I have been working on a similar sheet (which includes some configurations I was unaware of and have yet to confirm)... You can find it (for little while) here:

http://www.absomagic.com/magicsquareconfigurations.pdf

Will update as I get more information...

Andrew

Andrew,

First, thanks for your post. Second, I checked some of my magic squares and unfortunately did encounter some that didn't match. For example #15, and #16.
I believe there are a few different methods that you can use to calculate the squares. Are you using any particular one?

John

John:

I based my configurations on basically three sources: Doug Dyment's Stimulacra (a great little book), Chuck Hickok's Diagonal Magic Square (ditto, and uses the same configurations and the Spanish source cited above.

I typically use Chuck's method and have found that 15 & 16 do not always match depending on the calculations. I have not yet checked the later configurations (29-36) to see if they can be relied on (or whether they are, in fact, valid). According to his book (and Doug's) those *should* be valid, but in practice I have found that this is not always the case...

I have found that with the method I use in performances the left and right edge squares usually work (top and bottom) and often the top and bottom edge squares do as well. That being said, I often omit them in case the numbers used do not add up...

Additionally, 33 through 36 are suppositions of mine (need to be checked) and are solely based on the information provided in the Spanish PDF above (which I reproduce as 29 through 32...

I haven't checked with my friend Harry Lorayne to see if he has experienced these with his traditional method for construction.

It is likely (and I am waiting to hear from Scott Cram on this) that in "true" magic squares all configurations should work, but that in magic presentations a majority is significant enough...

What version do you use?

Andrew

(Do we know each other? I noticed NH in your signature...)

Andrew, no we have never met but do share the great state of New Hampshire.

I also have Doug's book, Scott Cram's articles, and Harry Lorayne's book. I believe I use Harry's method the most but I'll double check on which one I used when I posted the comment.

I am also cautious in talking about all the different ways they match because it seems there always are exceptions. Although I must admit, your number of possible solutions is the most I have seen to date!

I'm not familiar with Chuck Hickok's method. By the way, I love all mathemagic effects and use a variety of them.

John

Andrew observed
Different magic squares have different patterns; not all magic squares conform to the same "rules". In order to be considered a magic square, the rows, columns, and main diagonals must sum to the same number. Beyond that is a function of the method.

The chart in my Stimulacra book corresponds to the methodology described in that book. The methodology used in my Mindsights book is slightly different (because the performance goal is different), and so the corresponding chart differs from the one in Stimulacra.

Some magic squares (those in which the target number is divisible by four, for example) can have more combinations than others. It's the nature of the math.

Let me preface this by saying: I'm an idiot.

I apologize to Doug profusely for inferring that his charts applied across the board... I neglected to consider that not all configurations are valid for all the formulations. End result: my enthusiasm outran my comprehension. This fact goes a long way to explain why I have encountered varying degrees of success when I explain the permutations (and further explains why I need to understand the underlying principles.

Doug is being very gracious by letting me off the hook (and by patiently explaining where I went wrong).

I am a huge fan of Doug's and highly recommend his books (I feature at least one of his pieces regularly in my professional work). He is a gentleman and a scholar!

[Sorry Doug... don't hate me too much! ;O)]

Now, I want to understand the underlying math... I've done a fair amount of research on this (mostly associated with performance), but it is obvious now: time to get back to the books!

ajp

No apology necessary ... I just took it as (understandable) confusion. I merely wanted to make sure that others weren't misled.

There are many magic square offerings out there (albeit many reinventing an often-spun wheel), and it's easy to get the various approaches confused. The underlying math on all of them is fundamentally the same, but there are many subtle tweaks and other considerations that are important to some of us. Plus, of course, different presentational approaches.

Sleightly, that link no longer works. Do you have an update? Thank you.

You can make a chart in Excel easily.

Here's a page from my Idiopraxis book, illustrating the basic combinations. There are a few more combinations, but they are not symmetrical, so I would never reference them in a presentation.

Click here to view attached image.

I think it is time to update this thread because I can now provide a definitive answer. Doug Dyment had it right when he said that the number of patterns that will succeed depends upon the method of construction. That explains why sources will vary in their conclusions.

The answer can also vary depending upon what you mean by a pattern and whether or not you allow duplicate values in the square. As an extreme example, a 4-by-4 magic square consisting entirely of the same number will have every possible combination of 4 cells summing to the same value. That square would not be interesting, but it makes the point that duplicates matter.

Most people would agree that the patterns that anyone should care about are the symmetric ones. If three cells from the first row and one from the third row happen to add up to the magic sum, no one would see that as more than a random coincidence. Also, a magician using a magic square for a performance would want the patterns to be predictable, that is, to know even before starting which ones will succeed.

In what follows, I will be describing order-4 magic squares without duplicates.

1. Every magic square, no matter how it is constructed, will have fourteen patterns that succeed. In addition to the ten required by the definition (i.e., the 4 rows, 4 columns, and 2 diagonals), the four outer corner cells, the four center cells, the top and bottom center edge cells, and the left and right center edge cells will always add up to the magic sum.

2. Any decent construction method can always achieve ten more: the four quadrant squares (2-by-2 squares in the corners), the four spread squares (corners of 3-by-3 sub-squares), and the two even broken diagonals. That will give you 24 symmetric patterns that can always succeed.

3. With a little extra attention, you can always get twelve more unless other constraints apply (such as requiring certain values in certain cells). That brings the total to 36.

4. If the magic sum is even (and you know how and no external constraints interfere), you can always achieve 52 successful symmetric patterns, and know before you start what they are.

So, in summary, 52 is the maximum when the magic sum is even, and 36 is the maximum when the magic sum is odd. Some might argue that certain ones should not be counted because they are not commonly referenced, and of course, you would never want to take the time to verify every one in a performance. Still, it is nice to know that you can claim more than you verify. For example, you might want to hand out a business card with all of the successful patterns shown on the back.

How can I be sure of these results? I developed a general formula that covers all possible 4-by-4 magic squares. I was not the first, (Ernest Bergholt published one in 1910), but I think I am the first to get one that is symmetric, so that you can see the conditions under which the various patterns succeed. The details are in my recent book, Magic Square Methods and Tricks, available from www.magicbookshop.com.

If I remember right, Sam Dalal has a version which gives out 52 combos.

I am planning to check out this book too...

That is correct. Dalal published a book called Patterns of Perfection in 1993, which may have been the first to identify all 52 patterns. He improved his method in 2011, which you can find in an inexpensive e-book at www.lybrary.com. it is called Patterns of Perfection Revisited. As I say in my book, I respect his method, but I prefer my own. If you compare them, you can draw your own conclusion.

You can find the 52 patterns (even totals) and 36 patterns (odd totals) in my ebook "Mostly Perfect". Also the 24 "safe" patterns that work with all rotations and reversals.

http://www.lybrary.com/mostly-perfect-p-124662.html

My construction method is essentially the same as Sam Dalal's, though somewhat easier to follow (in my opinion).

I am looking forward to Jim's book. I have an idea how his method works and, if I am correct, it will be very elegant. I've been trying to crack this myself for some time.

Mike

Thanks for the confidence, Mike, even before you have even seen what I discovered. I believe you will be pleased with the elegance and versatility of the method.

I will also recommend your "Mostly Perfect" ebook to anyone who has not seen it. I did not want to expose methods that were not mine, so I did not explain all of the wonderful methods I found while doing my research. However, I included references to the ones I thought were worth study. Yours is certainly among them. I view all of these as adding to the stock of knowledge, rather than competing for the best method.